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For a nonzero reference transfer function $H_{\mathrm{ref}}$, the complex relative gain error is the relative errorIt measures the distance from the complex ratio to one, so it responds to both amplitude error and phase error. A vector vector network analyzer measures both quantities; replacing its complex result by a magnitude discards part of the measured information.^[ref]

Write $H/H_{\mathrm{ref}}=a{e}^{i\varphi }$, where $a\ge 0$ and $\varphi$ is the phase difference. Multiplying by the complex conjugate givesThus a pure five-degree phase error has $E=2\sin (2.5\pi /180)\approx 0.08724$, about $8.72\%$, even though the magnitude ratio is exactly one. For small $\delta =a-1$ and small $\varphi$ measured in radians, $E\approx \sqrt{{\delta }^2+{\varphi }^2}$. Adding the two unsigned errors is generally not the same operation.

The metric also controls signals. Suppose two linear time-invariant systems have frequency responses $H,H_{\mathrm{ref}}$ and act on the same input in L2 space. On its frequency support assume $H_{\mathrm{ref}}\ne 0$ almost everywhere and $|H/H_{\mathrm{ref}}-1|\le \epsilon$ almost everywhere. Applying the norm bound for a Fourier multiplier on L2 to the reference output yieldsThis is an output-relative bound, so input frequencies suppressed by a zero of $H_{\mathrm{ref}}$ require separate treatment. Near a reference zero, a small absolute error can produce an enormous complex relative error; an absolute or explicitly fixed-scale tolerance is then more informative.

A vector network analyzer measures complex reflection and transmission of electromagnetic signals as a function of frequency. The word vector records that both magnitude and relative phase are measured. It is used in electrical measurement of devices such as transmission lines, filters and amplifiers; the measurement is a comparison of outgoing signals with a reference excitation.

A primary manufacturer explanation describes a tuned receiver that down-converts the measured signal and compares it with a reference split from the source. Its test set samples the reference signal automatically. A longer signal path produces a frequency-dependent phase delay, even if its magnitude is unchanged.^[ref]

For a two-port device, scattering parameters relate incident wave amplitudes $a_1,a_2$ to outgoing amplitudes $b_1,b_2$:When port 2 has no incident wave, $S_{11}=b_1/a_1$ and $S_{21}=b_2/a_1$. The termination and reference-plane conditions belong to the definition. For equal real reference impedances with power-normalized waves, $|S_{21}{|}^2$ is the transmitted-to-incident power ratio under that condition.

For example, $S_{21}=0.5e^{-i\pi /3}$ has magnitude $0.5$, phase $-60^{\circ }$ and power ratio $0.25$. Its logarithmic magnitude is $20{\log }_{10}(0.5)\approx -6.02\mathrm{dB}$. A magnitude-only measurement loses the phase. A pure delay $\tau$ contributes $e^{-2\pi if\tau }$; a $2\mathrm{ns}$ delay changes phase by $-36^{\circ }$ over a $50\mathrm{MHz}$ frequency increase, without changing magnitude. This is an original numerical example of the manufacturer's reference-delay principle.

Instrument calibration and a stated reference plane are consequently important: connecting cables and fixtures can contribute their own response. Phase wraps also require care because angles differing by $360^{\circ }$ describe the same complex value. The complex relative gain error can quantify a deviation that would be invisible if only transmitted magnitude were compared.